Zak Transform: Theory & Applications

• Zak Transform is a representation that recasts functions into quasi-periodic structures over a fundamental cell, bridging the spatial and dual frequency domains.
• It exposes underlying signal structure by diagonalizing translation-like actions and preserving unitary properties across continuous and discrete settings.
• Its versatile applications span Gabor frame theory, delay–Doppler communications, quantum error correction, and noncommutative analysis, providing practical design insights.

The Zak transform is a representation that recasts a function or sequence as a quasi-periodic object on a fundamental cell, typically pairing a spatial or delay variable with a dual frequency or Doppler variable. In continuous settings it maps functions on $L^2(\mathbb{R})$ to quasi-periodic functions on a strip, and in discrete periodic settings it maps length-$MN$ sequences to $M\times N$ arrays. Across harmonic analysis, Gabor frame theory, invariant-subspace analysis, delay–Doppler communications, and Gottesman–Kitaev–Preskill coding, its central role is to expose periodic structure, diagonalize translation-like actions, and preserve Hilbert-space geometry (Kloos et al., 2013, Mehrotra et al., 30 Mar 2025, Barbieri et al., 2014, Pantaleoni et al., 2022).

1. Definitions and canonical realizations

In one continuous normalization, for $f:\mathbb{R}\to\mathbb{C}$ and $\alpha>0$, the Zak transform is

$$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$

whenever the series converges. The transform is completely determined by its values on the fundamental domain $[0,\alpha)\times[0,1/\alpha)$. A second common normalization writes

$$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$

and is used in the analysis of totally positive functions and their zero sets (Kloos et al., 2013, Kloos, 2014).

In the discrete periodic setting used in Zak-OTFS, time-domain signals are $MN$-periodic complex sequences $\mathbf{x}[n]$, and the discrete Zak transform is

$MN$0

For fixed $MN$1, $MN$2 is an $MN$3-point DFT of the polyphase component $MN$4. This converts a length-$MN$5 sequence into an $MN$6 delay–Doppler array (Mehrotra et al., 30 Mar 2025).

A finite Zak-transform formulation for a period-$MN$7 sequence with $MN$8 is

$MN$9

with inverse

$M\times N$0

This form is used to construct perfect and zero-correlation-zone sequence sets and to interpret the delay–Doppler grid in OTFS as a Zak domain (Peng et al., 9 Feb 2025).

2. Quasi-periodicity, basis structure, and unitarity

The defining structural feature of the Zak transform is quasi-periodicity. In the continuous case,

$M\times N$1

so the transform is periodic in the dual variable and quasi-periodic in the primal variable. In the discrete periodic case, the corresponding relation is

$M\times N$2

which makes the delay period quasi-periodic and the Doppler period strictly periodic (Kloos et al., 2013, Mehrotra et al., 30 Mar 2025).

In the discrete theory, the time-domain space is an $M\times N$3-dimensional Hilbert space with orthonormal basis $M\times N$4, indexed by $M\times N$5 and $M\times N$6, where each basis vector is a time-limited tone of length $M\times N$7 repeated periodically with period $M\times N$8. Applying the DZT yields quasi-periodic delay–Doppler arrays $M\times N$9, and the map $f:\mathbb{R}\to\mathbb{C}$0 sends one orthonormal basis to another. Consequently, the DZT is unitary and preserves inner products; in Zak-OTFS this is the mechanism by which correlation and ambiguity properties are transported from the time domain to the delay–Doppler domain (Mehrotra et al., 30 Mar 2025).

A closely related picture appears in the Zak basis used for bosonic quantum systems. The kets

$f:\mathbb{R}\to\mathbb{C}$1

satisfy

$f:\mathbb{R}\to\mathbb{C}$2

and furnish a continuous orthonormal basis on a Zak patch of area $f:\mathbb{R}\to\mathbb{C}$3. In that setting, the Zak transform is literally the wavefunction in the $f:\mathbb{R}\to\mathbb{C}$4 basis (Pantaleoni et al., 2022).

3. Gabor analysis, frame theory, and zero sets

In Gabor analysis, the Zak transform gives explicit criteria for frame bounds and frame failure. For $f:\mathbb{R}\to\mathbb{C}$5, rational densities $f:\mathbb{R}\to\mathbb{C}$6 admit frame-bound descriptions in terms of matrix-valued Zak transforms. In the case $f:\mathbb{R}\to\mathbb{C}$7, $f:\mathbb{R}\to\mathbb{C}$8, the optimal bounds are

$f:\mathbb{R}\to\mathbb{C}$9

At critical density $\alpha>0$0, the Balian–Low theorem uses the quasi-periodicity of $\alpha>0$1 to show that a continuous Zak transform cannot be non-zero everywhere on its fundamental domain (Kloos et al., 2013).

A major line of work concerns the zero sets of Zak transforms of totally positive functions and exponential B-splines. For periodic exponential B-splines $\alpha>0$2 of order $\alpha>0$3, $\alpha>0$4 has exactly one zero in $\alpha>0$5, located on the line $\alpha>0$6. Via the explicit relation

$\alpha>0$7

the same single-zero property transfers to finite-type totally positive functions, and the frame set for such windows is the full subcritical region $\alpha>0$8 (Kloos et al., 2013).

For totally positive functions without Gaussian factor in the Fourier transform, the finite-type picture extends to infinite type. The complexified Zak transform is holomorphic in a strip, the finite-type approximants converge uniformly there, and Hurwitz’s theorem is used to show a zero-free strip away from $\alpha>0$9. The resulting theorem states that there exists $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$0 such that

$$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$1

so the Zak transform again has exactly one zero in the fundamental domain (Kloos, 2014).

4. Generalizations to group actions and noncommutative settings

The Zak transform extends far beyond $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$2 and $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$3. For a discrete countable LCA group $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$4 acting measurably and quasi-$$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$5-invariantly on a $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$6-finite measure space $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$7, with associated unitary representation $$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$8, the generalized Zak transform is

$$Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega}, \qquad (x,\omega)\in\mathbb{R}^2,$$9

It defines an isometric isomorphism

$[0,\alpha)\times[0,1/\alpha)$0

where $[0,\alpha)\times[0,1/\alpha)$1 is a tiling set for the action, and it satisfies the covariance relation

$[0,\alpha)\times[0,1/\alpha)$2

This diagonalization converts $[0,\alpha)\times[0,1/\alpha)$3-invariant subspaces into multiplicatively invariant spaces and yields a range-function classification of invariant subspaces, frames, and Riesz sequences (Barbieri et al., 2014).

A semidirect-product version is available for $[0,\alpha)\times[0,1/\alpha)$4, where $[0,\alpha)\times[0,1/\alpha)$5 is locally compact, $[0,\alpha)\times[0,1/\alpha)$6 is LCA, and $[0,\alpha)\times[0,1/\alpha)$7 is a $[0,\alpha)\times[0,1/\alpha)$8-invariant uniform lattice. The transform

$[0,\alpha)\times[0,1/\alpha)$9

maps $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$0 into a Zak space on

$$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$1

and satisfies the Plancherel identity

$$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$2

This recovers the classical transform when $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$3 is trivial and supplies explicit realizations for groups such as $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$4 and Weyl–Heisenberg groups (Farashahi et al., 2012).

A different noncommutative extension arises from the Weyl transform on $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$5. For $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$6 with Weyl kernel $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$7, the Weyl–Zak transform is

$$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$8

a unitary map into $$Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},$$9. It diagonalizes twisted translations: $MN$0 which leads to bracket maps and frame, Riesz-sequence, and Schauder-basis criteria for twisted shift-invariant spaces (Ramakrishnan et al., 2023).

5. Delay–Doppler communications, waveform design, and OTFS

In Zak-OTFS, the transform provides the defining map between time-domain signals and the delay–Doppler lattice. A delay–Doppler impulse at $MN$1 is mapped by the inverse discrete Zak transform to the time-domain “pulsone”

$MN$2

and the transmit waveform is the superposition

$MN$3

In this framework, the Zak-OTFS carrier is a pulse in the delay–Doppler domain, and the Zak transform converts it to a pulse train modulated by a tone in the time domain (Mattu et al., 10 Aug 2025, Mehrotra et al., 12 May 2025).

The discrete Zak transform preserves inner products, and the paper “Zak-OTFS for Mutually Unbiased Sensing and Communication” shows that time-domain CAZAC sequences determine quasi-periodic delay–Doppler arrays with highly structured ambiguity functions. For the quadratic-phase CAZAC family, the self-ambiguity in the Zak domain is supported on the discrete line

$MN$4

and for two distinct family members the cross-ambiguity magnitude is constant: $MN$5 The same work shows that these waveforms are mutually unbiased with respect to every Zak-OTFS carrier and are suited to integrated sensing and communication and to 2-step RACH preambles (Mehrotra et al., 30 Mar 2025).

Several recent developments use additional unitary transforms on top of the Zak basis. “Zak-OTFS with Spread Carrier Waveforms” constructs an orthonormal basis of spread carrier waveforms with low PAPR, realized by a unitary transform based on the discrete affine Fourier transform; the proposed spread carrier-based Zak-OTFS achieves full spectral efficiency like pulsone-based Zak-OTFS, with $MN$6 dB lower PAPR per basis element and low PAPR only $MN$7 dB (Mehrotra et al., 12 May 2025). “Low-Complexity Equalization of Zak-OTFS in the Frequency Domain” derives a frequency-domain system model unitarily equivalent to the delay–Doppler model, shows that the frequency-domain channel matrix is banded, and obtains equalization complexity linear in the dimension of a Zak-OTFS frame, in contrast to cubic complexity for naive MMSE equalization (Mattu et al., 10 Aug 2025).

The Zak framework also underlies practical pulse-shaping and over-the-air implementations. In discrete oversampled Zak-OTFS, every delay–Doppler domain symbol undergoes the same effective channel response, and the I/O relation remains a twisted convolution; analysis of ambiguity functions shows that high sidelobes widen channel spreading, motivating a PSWF/IOTA pulse design with superior channel estimation accuracy and BER in the high-SNR regime (Zhang et al., 7 Feb 2026). An over-the-air mmWave demonstration implements Zak-OTFS with root-raised-cosine filtering, modulations up to 16-QAM, and a low-overhead preamble, and develops a signal model in which carrier-frequency offset and timing impairments are jointly captured within the effective DD-domain channel (Ramachandran et al., 10 Nov 2025).

6. Quantum, operator-theoretic, and modern analytical perspectives

The Zak transform furnishes a particularly compact description of the square GKP code. Choosing Zak period $MN$8, the ideal GKP codewords

$MN$9

collapse in the Zak basis to

$\mathbf{x}[n]$0

The syndrome projector for modular displacement $\mathbf{x}[n]$1 becomes

$\mathbf{x}[n]$2

and the error-corrected logical amplitudes are samples of the Zak transform,

$\mathbf{x}[n]$3

This leads to a modular-variable subsystem decomposition in which a single bosonic mode is expressed as a virtual qubit tensored with a virtual gauge mode, and tracing over the gauge mode yields the logical state associated with the oscillator state (Pantaleoni et al., 2022).

A complementary analytical development characterizes function and distribution spaces directly in terms of Zak-transform estimates. For an ordered basis $\mathbf{x}[n]$4, $\mathbf{x}[n]$5 is a homeomorphism from $\mathbf{x}[n]$6 onto the space of smooth quasi-periodic functions of order $\mathbf{x}[n]$7, and it extends uniquely to a homeomorphism on tempered distributions. The same program gives characterizations of Gelfand–Shilov spaces, their duals, and modulation spaces by Wiener-type estimates of Zak transforms, and it establishes necessary and sufficient conditions for linear operators to be conjugations by the Zak transform (Toft, 2017).

Taken together, these developments place the Zak transform at the intersection of quasi-periodic harmonic analysis, finite-dimensional signal design, group representations, delay–Doppler communications, and bosonic quantum information. A consistent theme is that periodicity in one domain and localization in a dual domain are not merely represented by the transform; they are reorganized into a geometry in which basis structure, ambiguity, frame bounds, invariant subspaces, and correctable errors become explicit.